Deployable structures inspired by the origami art

Deployable Structures Inspired by the Origami Art

by

Ken Giesecke

Bachelor of Arts, University of Pennsylvania, 1998

Submitted to the Department of Architecture in partial fulfillment of the requirements for the degree of

Master of Architecture

at the Massachusetts Institute of Technology February 2004

@2004 Ken Giesecke. All rights reserved.

The author hereby grants to MIT permission to reproduce and to distribute

publicly paper and electroric copes of this thesis document in whole or in part.

signature of author:_ _{Ken Giesecke}

Department of Architecture January 16. 2004 certified by:_ Assistant Professor, John Ochsendorf Department of Architecture Thesis Supervisor MASSACHUSEITS INST11~UTE OF TECHNOLOGY

FEB 2 7 2004

LIBRARIES

Carol Burns Visiting Associate Professor, Department of Architecture

Thesis Supervisor

Bill Hubbard, Jr.

Adjunct Associate Professor of Architecture

Chairman, Department Committee on Graduate Students

ROTCH accepted by:

thesis committee

Thesis Supervisors John Ochsendorf

Assistant Professor, Department of Architecture

2 Carol Burns

Visiting Associate Professor, Department of Architecture

Thesis Readers

Shun Kanda

Senior Lecturer, Dept. of Architecture

Martin Demaine

abstract

Deployable Structures Inspired by the Origami Art

by Ken Giesecke

Submitted to the Department of Architecture in partial fulfillment of the requirements for the degree of Master of Architecture at the Massachusetts Institute of Technology

February 2004

My thesis is an exploration of design methods and tools using origami as a vehicle to test their usefulness

and coming to terms with their limitations. I have taken my fascination with a particular development in origami and put my belief in its potential for architectural application to the test by way of various investiga-tions: materials and structural analysis, mathematical reasoning, manipulating space and form, parametric modeling, fabrication, and finite element testing.

Parting from conventional, figural forms, mathematicians developed open-surface forms together with theo-rems that governed the ability of these folded forms to fold flat. I selected a particular form, the Kao-fold,

for its simplicity, beauty, and structural properties and imagined many exciting possibilities, specifically for its application in designing a deployable structure.

I analyzed its crease pattern, exploring variations and their corresponding folded forms.

Simultaneously, different material ideas for larger-scale structures were tested and a particular configuration

was assessed for internal stresses and its structural stability. Its transformation from a flat sheet to a folded state was scrutinized under the lens of mathematical reasoning, namely trigonometry, by linking the acute angle of its crease pattern and the dihedral angle in its folded state to its final folded configuration. The rigidity of this investigation was offset by the freedom afforded in manipulating paper models. As such, different spatial qualities and forms were explored while addressing the issue of scale and potential

applications. The transformational characteristics discovered were digitally simulated via the construction of

parametric models, which was a more controlled manipulation of the form in a virtual space. In order to go beyond the realm of representation and address real-life building issues, a temporary open-air shelter was

designed and constructed in detail. The goal was to tackle the complexity of assigning materials, designing components and fabricated them. As a final endeavor, the model's construction was tested for its structural stability using a finite element software.

Thesis Supervisors: John Ochsendorf Title: Assistant Professor, Department of Architecture

to my advisor, John Ochsendorf, for the consistent enthusiasm he brought to every discussion. It has helped me tremendously to stay guided and

motivated

acknowledgement

to Carol, my co-advisor, for addressing my assumptions and my shortcom-ings

to my readers, Shun Kanda and Martin Demaine

to my unofficial reader, Erik Demaine. Your interview in the New Scientist led me to pursue applications of folding in a seminar which paved the way to my thesis

4 _{to professor Kenneth Kao who gave the seminar and for teaching us to}

think critically

to Alex for generously giving me so much of his time and energy toward

realizing the "sphere model"

to Susan and Victor for showing me the ropes to Catia

to other members of the "quiet studio" - Mike, Karim, Chris, Aliki and Tracy

for their moral support

table of contents

03 06 08 10 14 20 abstract

motivation for research precedent studies

origami: investigating open surface forms

addressing materials: a preliminary structural analysis a mathematical investigation:

comprehending the transformation by trigonometry paper models: a physical manipulation of space & form

parametric modeling: digital manipulation through defining parameters fabrication: beyond diagrams & representations

finite element analysis: an assessment of structural stability conclusion

comments & criticisms bibliography

motivation

for research

I developed my interest in deployable

struc-tures and origami in a seminar taken in the

semester preceding my thesis at the

Gradu-ate School of Design in Harvard taught by

professor Kenneth Kao titled GSD 6410 Con-cept to Construction: Building Components & Assemblies. The course examined and dis-cussed various case studies of instances in architecture where truly innovative thinking in the engineering of a structural component

enabled unique design solutions to emerge. The work of engineers/designers such as

6 Jean Prouvet, Buckminster Fuller, and Peter Rice were examined for their inventiveness.

In this context, students were taught to think critically about current-day design and

building technologies and try to bring new insight into their use as a starting point for innovation. Eventually, this led to the final assignment for which I researched

collaps-ible furniture and structures and in parallel

explored basic folding patterns in origami. The research was guided by the belief in

origami to be able to inform the design of

deployable structures and focused on the design possibilities of a particular pleated

form (which I eventually named the Kao-fold as no inventor could be identified).

precedent

studies

The ideas and thoughts

sur-rounding some of the projects

discussed in the course as well

as those researched for my final assignment have carried

8 over into the development of

my thesis. Certain precedent

studies that have been more inspiring may be presented:

A geodesic dome uses a pattern

of self-bracing triangles in a pattern that gives maximum structural advantage, thus

theo-retically using the least material possible. (A 'geodesic" line on a sphere is the shortest distance between any two points.)

In Renzo Piano's IBM pavilion, a series of polycarbonate pyramids are reinforced with cast aluminum connections spliced and glued into laminated beechwood.

( 6.

Commissioned by Rice Univer-sity Art Gallery, Bamboo Roof is an outdoor site-specific installa-tion created by Japanese Archi-tect Shigeru Ban.

The work features an expansive open-weave canopy of bamboo boards that spans the Gallery Plaza.

Metallic

frarin

Metalic angle

Blow oint

joint details for an accordion roof

Water-tight hinges drive the design in a

motor-driven aluminum sheet roof.

(7.)

,

₄

origami: investigating

open-surface forms

Starting in the 1930's, the art of origami began to extend beyond traditional, figural works with which we most strongly associate with origami. Liapi, 386 Enthusiasts' devel-oped an interest in the mathematical

proper-ties found within the crease patterns. The crease pattern is literally the pattern of creases seen in the paper once the origami

form is unfolded into a flat sheet. (see examples shown to the right)

Thus, working backwards, one is able to

10 determine the final form for a particular design once the crease pattern is known.

The only key required to unlocking this DNA which determines a certain form is to know

whether each individual crease within the

pattern results in a mountain or a valley fold.

If either side of the crease is folded towards

you, that is a valley fold and if away, you get

a mountain fold. (see image on bottom-right)

(8.) (8a.) -/ ( (9

Biaxial Flasher Supreme by Ushio Ikegami.

The thicker lines represent mountain folds.

Cubic Flahser by Ushio Ikegami.

Extremely dificult to fold, the cubic flasher does not expand and con-tract smoothily like a basic flasher.

In the crease pattem shown,

divid-ing a square into four equal parts, all the creases represent valley folds.

One category of origami, which developed out of this newfound interest in origami's

mathematical properties is modular or unit origami. Unit origami, as it's name describes,

is formed from a number of repeating units

which link to form various closed shapes, which in turn, may be linked to one another with certain folded units. (see image)

A second development was open-surface

forms. These open-surface forms had cap-tured my interest, as they seemed to carry great potential for being applied as architec-tural elements, be it an outer skin of a build-ing, an interior surface or an entire habitable

structure. Together with the development of

theorems which governed the ability of these

open-surface forms to be folded flat, one may

imagine exciting possibilities for origami's

application in designing deployable

struc-tures - strucstruc-tures that could be collapsed,

transported, and efficiently deployed.

These theorems were derived by mathe-maticians through a careful understanding

for how the crease pattern of the pleated

form could be manipulated to give specific resulting forms. They determine the underly-ing mathematical properties of folded forms which allowed them to fold flat. Liapi, 387 These topological conditions that any pattern

of creases must fulfill are:

The 6-unit wedge is

formed by a series of modular belt-units.

(10.)

a. At any vertex, the sum of the angles is 360 degrees

b. The number of creases originating at a vertex must always be even. c. (the Hushimi theorem) When four creases meet at a common point, the difference between two adjacent angles has to be equal to the difference

between the remaining two angles.

d. (the Kawasaki theorem) The angles al, a2, a3 ...., a2n, surrounding a single

vertex in a flat origami crease pattern must satisfy the following requirement:

al + a3 + a5 + ... + a2n-1 = 180 degrees

and

One such open-surface form

that was of particular interest

was the Mars fold, patented

by Taborda Barreto. The Mars

fold is composed of squares and rhombuses, lying adjacent to the squares. What was so

fascinating about the Mars fold

was how its simple crease pat-tern could yield such a variety

of forms depending on the direction of the folds. Yet, the problem with these

forms was that they often required large amounts of

12 bending and twisting both to

develop the form and to keep it in its folded state. If the

struc-ture were to be constructed on

a larger scale, a material other

than paper would not be as

for-giving.

While the Mars fold and other

open-surface forms could be shaped without any distortion within the surface it was also vital that one could give the

pleated surface a degree of

cur-vature.

The red and blue lines represent mountain folds and the green, valley folds, Variations in the Mars fold are based on two pos-sible positions for the rhombuses. While all patterns fold flat, the only restriction is that the lines forming each obtuse angle of the rhombs be folded in the same direction.

This second characteristic (which can not be found in the fishbone or tessellated stars fold) was important in being

able to contain space and offer

a sense of enclosure. This

cur-vature could be implemented into the pineapple fold,

how-ever, it would not translate into

an efficient structure as there is

a significant amount of

overlap-ping between the panels. Ultimately, I returned to the Kao-fold for its versatility, beauty, but most importantly, its inherent structural value.

Among other open-surface 13

forms, it has a clearly identifi-able load path and a

consid-erable depth to its section. It

reminds one of a truss, whereby the greater the depth of the sec-tion, the stronger it becomes and the greater the distance

it may span. However, as

the form is not composed of linear elements, differing from the truss, the panels offer cov-erage and allow one to imagine

addressing

materials:

a preliminary structural

analysis

After playing with different configurations of the Kao fold in paper, I was eager

to work with materials besides paper that would begin to suggest the difficulties of working on a larger scale. Different material/

assembly ideas were tried with the goal of

keeping the paper model's ability to expand and collapse freely. In looking at different material possibilities, it soon became

appar-14 ent that paper could not be substituted with

a single sheet material but that it would be necessary to make a composite where the

material along the crease lines of the paper model would be flexible differing from the rest of the form that could be rigid. For this

flexible material along the fold-lines, drywall

and vinyl tape were tested. In the panels,

chipboard, acrylic, and thin plywood were

tried. The elastic vinyl tape was used for,

in order to fold the structure, the material

would need to be stretched at the nodes. As can be seen in the chipboard and dry-wall tape model, another solution was to

simply extract material at the nodes thereby disconnecting one seam from the next.

40

DEVELOPED SHAPE

FOLDED SHAPE

(11.)

The chip model formed by two 90 degree modules joined at their ends was part of a study by C.G. Foster with S. Krishnakumar in 1986 of a family of foldable, por-table structures based on triangu-lar panels.

I assigned the Kao fold the form

of a half-arch as a starting point

to understand how the stresses from its own dead weight are carried through the structure to the ground. The making of a half-arch chip model using

zip-lock bags (polyethylene) along

the seams was particularly

instructive in being able to see

how the stresses caused by the materials own dead weight were carried through from one panel to the next, that is, along the seams.

A heavier plywood model was

16 made to increase the stresses

on the seams and thereby

clar-ify exactly how the seams were resisting the dead load of the

panels. Furthermore, given that the panels are much larger than the chip model, the connections

between them and the seams could be elaborated. Grooves were routed out so that an

inter-stitial strip of wood could be sandwiched between two layers

and hold the sheet material along the entire edge.

The transfer of weight from the top of the arch to the base could be abstracted using short steel rods and rope. From this

abstraction, it becomes clearer

as to how the internal stresses

inside the panels are operating,

that is, in compression along

the fold and in tension on the

other two sides. (see sketch)

The steel models, moreover, are instructive in identifying the

Kao-fold's weaknesses as a

structure. In the shape of a

half-arch, a tremendous moment

force exists at the base.

Fur-thermore, one can observe that, 17

laterally, it is very unstable (as it would sway quite freely with a slight force to the side!)

A first reaction to increase

lateral stability would be to complete the arch, thereby

transmitting the gravitational forces from one end to the other. The model was

con-structed using two half-arches connected by a pin joint at the ends.

One means to stabilize the structure in the lateral direction with external sup-ports would be to have opposing ten-sion forces applied to it, in the way a young tree is harnessed to the ground.

In observing the internal forces of another abstract model of a slender

steel rod held in tension by a string,

another consideration was to use

embedded pre-tensioned cables which would force the structure to erect itself while counteracting that force with

external tension cables running diago-nally for lateral stability, from the top of

the arch to the ground.

18 _{In accounting for the moment force at}

the base, however, it would be a more

stable structure if the embedded

ten-sion cables were brought on the out-side, again running diagonally, and, for sake of elegance, the counteracting cables be placed along the inside

sur-face. ... ...

A Mathematical

Investigation:

comprehending

the transformation

by trigonometry

Simultaneous to thinking about materials and structural stability, I wanted to come to terms

with the mathematical principles

20 behind the transformation of the Kao fold. That is, to determine the relationship between the

geometry of the crease pattern

and its corresponding folded form.

After playing with paper models it became apparent that the size of the acute angle in the crease

pattern controlled the degree of

curvature in the pleated arch.

The smaller the acute angle, the

"tighter" the curl of arch would

Furthermore, in looking at the section, it

may be deduced that in order for a half- -4-0"

arch to stand perpendicular to the ground plane (In this state, the bottom segment of the arch is perpendicular to the ground plane and the top segment parallel to it. Also the vertices of the half-arch are tangent to a circle), the acute angles would need to add up to 90

degrees.

As the image shows, the smaller the acute angle is, the greater are the number of

folds. As a completed arch, structurally,

the Kao-fold behaves much like a truss in

a half-circle formation. (see image 12) 5.625

Just as "increasing the modular

incre-mentation of the truss decreases the

abil-ity of the vertices to resist concentrated 21

loads", the more faceted the surface of the Kao-fold, it loses depth compromising its strength. Pearce, p.27 Thus based on an intuitive assessment of the stability of the structure in accordance with its depth,

it was determined that the 11.25-degree

model would be developed.

*The size of the arched structure is dependent on the length of the seams. Each arch shown in the image was sized to house 2-3 people.

It is important to note that the study only assesses the impact of the

acute angles on the contained space seen in section. That is, there is no consideration for the impact of the dihedral angle. As the dihedral angle would affect the curvature of

the arch as well, its mathematical relationship had to be determined. In other words, the perpendicular condition achieved previously was

based on the assumption that the form was folded flat and, as such,

should the arch be deployed in a semi-unfolded state, the effect of

the dihedral angle would need to

22 be accounted for. By expanding and

collapsing a paper model, we learn that as the arch folds flat, it acquires a tighter curvature and vice-versa.

Intuitively, one may presume that to compensate for this affect, in order

to maintain a semi-unfolded

perpen-dicular state, the acute angle of the crease pattern must be increased

from 11.25 degrees to some greater

number. Through some basic trig-onometry we find that, should the dihedral angle be set at 90 degrees, this new value for the acute angle is

15.7 degrees. (see page 20)

a--V -.

\y B2 B A tan 11.250 = x/I r

T

7

tan 11.250 = x/36" x = 7.161" tan 0= yll tan 0 = 10.13/36" = 15.7*

The following is a way to calculate the acute angle, given that all units in the folded piece are in the same folded state i.e. 60 degrees or 90 degrees, so that the bottom segment of the arch is

perpendicular to the ground plane and the top segment parallel to it.

It has been previously deduced that the acute angle must be a factor of 90 ( 22.5, 11.25,

etc.) to allow for this condition. However, these calculations depend on the arch being in a fully

folded/closed state.

The less folded the arch, the greater its tendency to be "less arch like" and become a straight, flat form. In other words, the arch "rises", losing its curvature.

Therefore, in order to compensate for this effect, the acute angle would need to be slightly greater than say...22.5 Through some basic trigonometry the problem may be solved allowing for the

design of arches with various curvatures while controlling the degree of folding.

cos 45* = xly cos 450 = 7.161"/y

...

L--- C --- R

For example, lets say, in the above folded pair of triangles, the acute angle RLB is 11.25 degrees when the triangles are completely folded and the length of the folded edge is determined. In its closed state, we draw a line from point B to the midpoint of LR or C which gives

us a right triangle LCB.

Next we find the length of BC. (In its closed state, we can draw a line through B and C which is perpendicular to LR.) Then we open the fold by a determined angle, say 60 degrees. The diagram below shows the fold in section.

B2 B A

\ / \ I

C

Using the value of BC, we can calculate B2C the "new" length of BC once the fold is opened 60 degrees. Then we work backwards and calculate

the acute angle given our "new" length of BC or B2C.

The illustration on p.20 shows the two

corresponding crease patterns with the

acute angles of 11.25 and 15.7 degrees as well as their completely folded state. When completely folded we can see the

impact of the acute angle. That is, the higher the value, the tighter the curva-ture, and thus, if we were to unfold and

expand the arch by setting the dihedral angle at 90, the arch opens up and

rises to the aforementioned perpendicu-lar condition. This condition was

mod-eled by inserting square pieces of chip into the center of every folded unit

dem-onstrating the logic and clarity of math-ematical thinking.

24

To make things more interesting, I

col-lapsed one end of the half-arch

con-figuration. The result is a form that resembles an orange peel as the first curvature along the length of the arch is supplemented by a second curvature

along the arch's width. In this model, essentially, what is happening is that the dihedral angles are diminishing from one end of the arch to the other. Using the same thought process and

trigono-metric calculations as before, one may determine the value for each dihedral

angle in a semi-unfolded perpendicular

The 11.25-degree, perpendicular condition dictates that in a half arch, there will be three full folded panels and two, half folded panels.

In the image for the crease pattern, the arch extends an additional 25%, giving four full folded panels. Once the smallest dihedral angle is determined, the difference between its corresponding acute angle and the next pair is established. (If one wants to work with the

11.25-degree, perpendicular condition, the smallest acute angle cannot be less than 11.25 degrees, as the planes con not close in on

themselves!) In this design, I chose 1.636 degrees. Once all the acute angles are determined, their corresponding dihedral angle is

calculated using the same, right-angle principles as before but going backwards. That is why it is important that the acute angles are paired, that is, to be able to use the same method as before.

59.358' 4.688* 93.356* 8.) 11.26" - 2.344* x2 - 4.688* 7.) 12.896* - 29.679* x2 - 59.358* 6.) 12.896" - 29.679* x2 -59.358" 5.) 14.532" - 39.88* x2 --- 79.76" 4.) 14.532" - 39.88* x2 -79.76* 3.) 16.168* -46.678* x2 - 93.356* 2.) 16.168" - 46.678* x2 - 93.356* 1.) 17.804" - 51.727* x2 - 103.454 -1.) 17.804" -51.727* x2 -103.454 -2.) 19.44* -55.695 x2 - 111.39" 1 I

acute angle dihedral angle T

new Y value based on

decreasing length of changing length of

dihedral angle dihedral angle $

tan 11.260* = y136' ->y= 7167" - cos $= 7.161'17.167"--> 0= 2.344*

tan 12.896* y/35.781' ->y 81 92" ->cos 7.117"/8.192" -> =29.679*

tan 14.532* = y135.531" -> y 9.21' -> cos 4 = 7.067"/9.21' -> $ = 39.88*

tan 16.168* = y/35.258" -> y = 10.222" -> cos p = 7.013"/10.222" ->V = 46.678*

tan 17.804* = y/34.945" -> y= 11.222" -> cos , = 6.951"/11.222" -> q = 51.727*

tan 19.440" =y/34.617" ->y= 12.218" -- cos $ =6.886"/12.218' ->) = 55.695"

tan 11.25"= x/36 -- x 7.161" tan 11.25*= x/35.781" -> x = 7.117" tan 11.25*= x/35.531' -> x = 7.067" tan 11.25'= x035.258" -- x = 7.013" tan 11.25"= x/34.945" -> x = 6.951" tan 11.25"= x/34.617' -> x = 6.886" 103.454' 11 39"

While this investigation allowed me to establish the relationship between the 2-dimensional

crease pattern and its corre-sponding folded form through mathematical reasoning, the precision required was limiting

and far removed from the freedom of working in paper models.

I had attempted to apply the math to the making of a 3d, digital model. Unfortunately, while in theory, the mathematical

reasoning is sound, due to rounding-off the values within the calculations, we see a tight fit between panels at the base and as we work our way towards the top of the arch, the marginal errors become accumulate and are progressively transparent.

paper models: a

physical manipulation

of space & form

in order to explore the habitational

qualities, addressing scale and ₂₇

potential applications, I returned to

the freedom found in working with paper models. In exploring forms, I looked first for design ideas using the basic crease pattern of the

Kao-fold. The first gesture to suggest

occupation would be to simply insert a scale figure. And so I took the

half-arch and orange-peel

configura-tions and modeled them as such.

Another idea would be to merge the

qualities of the full-arch configuration

and the orange-peel and collapse the full arch at the ends and link the

Subsequently, to go further, I identified specific

characteris-tics of the 2-dimensional crease pattern, namely that of the axis

-of the dihedral angle and length of unit. (By axis, what is referred

to are the lines along which every folded unit lies, collinear with the vertices of the dihedral angles.) The following diagrams show

variations of these two charac-teristics and, for some, their cor-responding folded forms. The

use of 1/8" scale figures

imme-diately gave the sense for

the habitational/spatial quality of

28 these forms.

(see opposite page)

2.000 -z000, -III I II I I I \II Al 2000* - 3.000*-3 50* - 4s00o---0-

2

4 0'1. 14 II 0~2~

OV

0'3

V-73 2.000* asor - / 2.500-~' , 0.4 T2"

014

0'4k

To address a certain scale of

construction and different appli-cations for the Kao-fold, col-lages were made as well. The

half-arch construction could be used for booths in a carnival

or craft fair, the full arch could

be a monumental entry-way. If the Kao-fold was constructed

out of steel, it could serve a more permanent function

much like Calder's out-door

sculptures, or if cast-in-place concrete, we may think of Felix Candela's entryway to the

Pabellon Music School, or the

(14) (14a)

parametric modeling:

digital manipulation through defining parameters

Parallel to my exploration of space and form by using the hands to manipulate paper models, a different quality of freedom was found in the digital manipulation of form. While playing in paper, a certain insight on the transformational qualities was gained,

that is, to understand exactly how the individual folded units move in relation to each other as they contract and expand. As such, it must be stressed that both investigations took place at the same time. The relationship was a two-way road as the thinking

applied to defining the Kao-fold in the virtual space of the computer also helped to

identify ways in which to develop the various physical models.

It was decided that the transformational qualities would be best modeled parametrically

using the parametric modeling software, Catia. Two parametric models that would address two different ways about thinking of the Kao-fold's transformation were

devel-32 _{oped. Interestingly, the limitations of the software forced me to focus my thoughts such} that they way in which the model was constructed would limit what transformations

would be possible in the end. While a limit, one may argue that this, specifically, is the power of parametric modeling -to construct a digital 3d model in accordance with certain parameters or inherent characteristics of that form.

Both models work with the Kao-fold in its half-arch configuration. The drawing of the

first model was based on assigning an axis to the dihedral angles and linking them to offset planes as well as to rotating planes. While the dimensions of the folded form

would be distorted, the intention of the model would be to play with the angle of the

dihedral angle axis and the distance between them to give morphings of the original geometry.

Model was drawn referring to 2-d sketches to give the 3-d coordinates of the corner

points, which subsequently allows one to link them giving the seams of the folds.

The following story board, illustrates the model's construction:

I Parallel planes offset from each other were

constructed.

5. Here we see all sketches in 3d space.

9. All nodes or extremities of the two-dimen-sional sketch are connected to give the trian-gulation of the pleated form.

4. Each rotational

plane (which is linked to the offset planes)

is associated with a

two-dimensional sketch. Catia operates between two

workbenches -that

of the sketch and

2. Extension lines were projected...

'W ' 641 '.. +' . a 4 0 's . 4..

3. ...which would function as a rotational axis

for the rotational planes.

the 3d model. In the sketch mode, points that would later give the 3-d coordinates for the vertices were drawn.

7. Every rotational plane represents an axis

for each dihedral angle. T .LR Z

j/T~

F' t. 4 !A ~-~+4 ~ -7

12. The outlines are filled to give panels of zero thickness. A h 33 ( I if i-i

1w

/ '44. A - . I - I. C. Ok i I *. + 4 'k " ' A o " 4 4 4 1 ' _

A

14. The first parameter, the distance between the offset planes, may be adjusted incremen-tally to expand the width of the half arch.

,4,0

15.

21. Next, the second parameter, the position

of the rotational planes may be adjusted.

34

19. ...and the camera angle may be changed

to give the impression that the structure is lying on its side.

51 -AIb@O~ftof &MAP

* i,~ ~17.

T Im. * IL

22. As with the offset planes, their assigned value may be controlled and incremental, fist closing in on one end...

,~p p.4 ~ ~ --->

23.

18. ...or the assigned offset values may be

randomized...

24. ...and then expanding

4

25. ...or random and chaotic 26. If we play with both parameters, more exciting forms can be given. 27.

35 . .... ...

The second model was con-ceived based on the premise

that the outer-most points of the

form lay on the surface of a sphere. (see image below) The parametric underpinning of

the model would be the function

that as the diameter of the

sphere increased, so did the dihedral angle of the folded

form. And as the diameter

approached infinity, the form

would flatten out into a single plane or, if you could bend

your imagination, a point on the

sphere.

brainstorming with Alex on the chalkboard...

1. I started by using the dimensions from a

determined 2-d crease pattern.

~iI

4;]

I

6. ...allowing me to get the intersections between the sphere and the planes.

11. Along the x-axis or the width of the form, each "collection" of points lie on a circle...

2. What was important about this particular pattern was that its outer

"nodes" lying in the same plane also lie on a circle and that its inner nodes lie on a smaller, concentric circle in the same plane.

7. Point definitions along the length of the

arch were drawn taking the assigned extre-mum as the origin on each circle.

~ ~.4 j

12. ..which taper around an axis running through the center of the sphere.

3. Before working in Catia, a new 2-d drawing to determine

the distance between the planes on which each and every dihedral angle lies had to be developed.

1 19

8. Each point would reference the location of

the vertices within the crease pattern.

9. Points were defined on either side of the extremum...

5. These planes were subsequently

trans-ferred to Catia...

W 1 Ap L4U4e

10. ... giving us a "patch" of points.

4*

#~A 4 . t

13. The distance between the points along the curve is identical for

each pair of points and taken from the determined 2-d crease pattern.

14. It was critical to fix the distance between the points so that the 'patch" or area defined by the points on the sphere's surface would stay the same...

Ai ~

ilti

/

U

16 Returning to the original radius 17. ...ines were drawn by connecting the points... 18. ...and filled to create panels,

22. We see the arch expand and unfold. 23. Finally, the radius is set to 1000ft, at which point the arch

has nearly folded flat.

A

N'

20. The final step would be to assign a parametric function to the radius of the sphere so that as it increased in size, the

"patch" or folded form would flatten. The original radius is set at 7'-10".

21. Subsequently, the assigned value for the

radius is raised to 100ft.

fabrication:

beyond

diagrams &

representations

While origami may serve as an extremely rich source of beautiful forms and while mathematical reasoning, paper models, and digital simulations address interesting and useful methods for design, the big question in my mind has

always been how to go beyond diagrams, and abstract representations? The

challenge would be to tackle the complexity of addressing materials, compo-nents, and fabrication, as well as other real-life issues, such as, how does this

thing get transported and deployed?

40 To begin thinking of building in reality and building a detailed model, the first

step would be to determine an application. Working within the anticipated time constraint, it was determined that the structure would be a small-scale,

open-air shelter for a temporary out-door event such as an arts & crafts fopen-air. The

ensuing criteria are that it be easily transportable and easily deployed. More

specifically it: (a.) should be lightweight, (b.) may fold up and fit in the back of a pick-up truck, (c.) would be intuitive and quick to assemble, that is, have

a small number of parts to assemble on site and assemble without the use

of a crane but the labor of a few people, and, (d.) should use non-precious

materials, preferably reconstituted or recycled. As a deployable structure, the

form is not manipulated or reconfigured on the site but will be brought there

as a prefabricated assembly whereby the form and its corresponding folding pattern would be predetermined after a virtual exploration in the computer.

To this end, fabrication would be based directly on files generated from the

computer model, all components manufactured for a particular configuration of the folded form.

41

The crease pattern and its form was chosen for its gentle

cur-vature which suggests a

struc-ture where the wall and roof are a continuous surface provid-ing an adequate sense of enclo-sure and separating one booth or function from another, yet to not be too enclosed, resulting

in large areas of non-habitable space. (Looking at the previous

collage of the half-arch configu-ration on page 30, we can see how much space is given over to "storage".) I had decided against the arched form as from

previ-42 ous paper models used to study

habitation possibilities and from

the simple materials

experimen-tation models, I determined that

the structure would be most

simply erected on site if the

seams were not put under

stress. One end of the shelter is considerably higher and out-stretched over the other for sake of orientation. It's form is simple allowing for easy

repeti-tion in a larger, open area such as a park or plaza.

Using the same assembly as the ply-wood model, the detail model recre-ates the Kao fold using honeycomb board with canvas at the seams.

A mockup construction using trash

bags demonstrates the basic

assem-bly of two sheets of fabric pinned to

and connecting the panels on either side. (see image to right) Honeycomb board was selected for its recycled

paper content and high strength to weight ratio. The honeycomb is such

an efficient construction that it is used in train bodies and airplane wings.

(see image15,15a)

To give the structure lateral stability,

a flexible yet relatively stiff

mem-brane would be required at the seams. As such, canvas was chosen as something very affordable and

available. To connect the canvas to

the honeycomb board, strips of 144#

YUPO, a recycable, tear-resistant, synthetic paper was used. Due to a short supply, 32nd" birch plywood

was substituted on the exterior. The canvas would be riveted to the

YUPO and birch strips, which, in

turn, would be secured by a bolt run-ning through the panel. (see sketch)

43

r

2

(15.)

Modular belts of different thicknesses were folded to test YUPO's flexibility.

Using the dimensions of the honey-comb board sold as a limitation, the detail model was

constructed at a scale of 3/8ths of the full scale. Furthermore, given material and time con-straints, the model focused on a portion of the

entire structure that can be seen highlighted on

the crease pattern and marked in red on the

chip-board model.

With this particular form, the structure

essen-tially behaves like an accordion requiring a sec-ondary armature to hold the panels in place.

This armature would take the form of

com-pressive struts spanning vertically between one vertex and the next, their lengths defining the respective location of the vertices.

Components that would receive the struts had

44 to be designed. Given the twist in the surface

of the overall structure, the individual pairs of panels and their respective struts do not align orthogonally. (see image-ortho) This means the receiving ends for the two struts meeting at a vertex would have slightly differing orientations

and can not function as a simple hinge (see imageprotoA). To accommodate for this differ-ence, one approach would be to insert a gasket like material to allow for the strut to hinge

later-ally as well as loclater-ally. (see image-protoB)

In this model, Tyvek was used at the seems

con-necting the panels (see mockup assembly on pre-vious page) and the hinge component is attached to an excess portion of fabric at the vertex using grommets. (see image-protoC)

image-ortho

image-protoA image-protoB

45

The problem with this method is that a lot of stress is placed on the pin that con-nects the hinge to the strut such that it may

easily fatigue and fail under the structure's

own dead weight. A different approach was adopted in which the hinge would connect

directly to the panels and allow the area of contact to pivot. (see image to the right)

In order to further rigidify the structure, the

hinging action would be limited to the struts

while the component would hold the two panels to which it was secured at a fixed

angle. This meant knowing each different angle for every vertex. Once the panels of the chip-board model were crudely held in

place with brass struts and epoxy and its 46 final shape resolved, the model was brought

to the 3d-digitizer for a 3d scan giving a

digital model in Rhinoceros. Subsequently, the 3d model was imported into Catia and

by bisecting a plane through the vertex, their respective angles could be determined.

As a third layer of structural reinforcement,

the panels forming the dihedral angles

would be locked into place using a

sec-ondary system of hinges.(see image-hingeA) These hinges are also bolted directly to the

panels. They are a simple hinge that may

be tightened once all hinges on the vertices,

together with the struts, are implemented together, and assembled.

15

All components were drawn in

AutoCAD and designed to be able to be laser-cut, glued together, and assembled.

610 13 14 17 12 920 1--E 19 21 47 6 900 ₍₄ 77.70 'I. a

'N

(4 86 0 72.10 * a ...

Also the YUPO strips and the canvas joints were drawn-up digitally including holes for all

bolts and rivets for precise

assembly. (see previous page) Cutting the canvas was partic-ularly challenging as it had

to be secured to a layer of chip board in order that it not catch on fire. Furthermore, the

dimensions of the laser bed led

to splicing separate pieces to

create a continuous joint.

While speed and precision were

certainly advantages in using the laser cutter, it may be

48 pointed out that the constraint

imposed by the dimension of the bed led to a large amount of wasted material.

1/8" thick, birch-faced plywood was cut in a single pass. Though cut-lines were prepared for laser cutting by nestling, a lot of material was wasted.

The following thumbnail images form a story board which highlight the design process of going back and forth between the digital and the physical. What I learned was a design process. The model

is a prototype, not an answer but the result of a chosen design path with the tools that were

given to me. While ease of deployment was a prime concern with which I began the model, this

objective was side-lined as the structural response

during deployment had not been given sufficient

attention. As a consequence of the hinges

break-ing under too much bendbreak-ing stress and forcbreak-ing them to delaminate, the components could not be secured alone and an extra pair of hands was often needed. Should the components have been machined by a 3-axis milling machine out of steel,

of course, this would not have happened. Yet, the ₄₉

limitations served to guide the design and should

the connections be redesigned, they would not be given a fixed angle but operate freely (as do the hinges on the interior) and the 3-dimensional loca-tion of each vertex would only be dependent on the length of the struts. This would take off a lot of pressure from the connections and enable one to lock each vertex in place starting at the base and working upwards. Overall, it was a very frustrating process to assemble the structure but what I learned was how often you lose a certain amounts of control along the way due to various

inaccuracies and overlooked difficulties and thus,

the result is an interpretation of the intentions with which I began.

..

finite element analysis:

an assessment of

structural stability

After constructing the detailed model, I wanted to assess

the dimensions of the components for structural stability as well as test the form for different material assemblies.

How would the structure hold up under its own weight if

constructed in full scale? Would it be able to withstand

lateral wind loads? Numerous simulations were run in the finite element analysis program, SAP 2000. Different mate-rials were tested under their own dead weight and various wind loads. In doing so, the thickness of the plates was 52 varied and the diameter of the struts adjusted.

Unfortunately, the honeycomb board could not be simu-lated and so the material that was closest in its properties, aluminum, was used. Moreover, given the time constraint, a flexible seam, as used in the prototype, could not be

mod-eled resulting in a rigid folded-plate structure. Neverthe-less, it was believed that testing the model for different

load conditions, albeit a different construction, would give legitimacy to its form.

I began by running a simulation for an aluminum structure

with 3/16th inch thick plates, and a 1"/.063" thick aluminum pipe to serve as the struts. (see: Analysis_1) The

displace-ment, measured at the highest point was negligible. Next, a

wind load of 30psf was distributed over its entire 337 sq.ft.

area giving a total of 10 kips force. (see: Analysis_2,2a)

Analysis_1: 3/16" thick alupanel -1/4"diaml.063"thick alupipe struts

Ti

Analysis_2a: 30psf wind load over 337 sq. ft. (10 kips) positive x -global Applied both to the concave and convex direction of the surface, the displace-ment along the axis of the wind load measure less than an inch.

The table lists displacement values only under its own dead load for an increasing plate thickness and a diminishing diameter of the struts. As expected, as the thickness of the plates was increased, the greater the deflection, however, the change was not linear given the greater rigidity obtained from the weld along the seams. Surprisingly, the struts could be reduced to a diameter of 1/8 of an inch and the displacement remains under half an inch.

disptaccnwnt. d;spbccm# tiC

dis plar ttiPJlt.

displacetncnt.

o1o~

-.0d16~

- 0431" - O52~J

312 thick alupanel - 1dia/.63"hic aiuip stut

1"thick alupane! - 1"diam/ 063"thic aiupipe stut 2" thick aluparne - 1"dam/.063"tick aiupp stuts

disp/acement: 3/16" thick alupanel - 1" diam 0633"th/ickI auipe sturnts

displacemnent: 3/16" thick alu panel - 1/2"diamn .06 3"thick a'u-pie struts

displacemnent: 3/16" thick alupanel - 1/4"diam .063"Thic alPupip- strut

dispiacemient: 3/6" thick a/upane 1/8"daim &5"thick aupipe4struts

-17 dIs.placement: 3/ 16" thick a/u panei - no struts

1672 dispiacemnent: 3/16" thick stee/ - no struts

0022" displacement: 172" concrete shiel

In running two separate sim-ulations for aluminum, while

the scale factor exaggerates the deflection, we see that the

structure would distort more uniformly with struts on the inte-rior as well.

Imagining a more permanent

application, stress diagrams

were obtained for a model with 3/16th inch thick steel plates showing that it is generally

stable but would first show signs of fatigue at the folds in

the base where the dead weight

54 accumulates.

Finally, displacement readings

were obtained for a full-arch

configuration using both steel

and concrete. They

demon-strate that, for either material, the Kao-fold is a very stable structure.

Analysis3: 3/16th" steel plate - stress diagram

Analysis_4: 1/8" concrete shell w/ -.0022" displacement at summit Analysis 6: 1/2" concrete shell w/ -.0022" displacement at summit

55

conclusion:

What I learned was a design process; a process of

ques-tioning through making. Though, at the onset, certain goals were stated and kept in mind, each step would inform the

next allowing the design to evolve as if by a will of its own.

Every investigation, every tool had offered certain insights,

yet due to various inaccuracies, the outcome was always a transfiguration of the original, or as mentioned, an inter-pretation of the intentions with which I began. The results address the creative potential of origami as a database of beautiful, sophisticated forms from which design ideas may

be conceived. Should the exploration be pursued further, I would like to test different pleated forms for material and

assembly ideas with a revised approach to detailing of 56 the connections and of transportation, deployment, and

dis-mantling mechanisms. For the structure's overall optimiza-tion, the geometry and mechanism for deployability should be tailored more specifically to its application.

Further future considerations include an evaluation of the

form for natural and artificial lighting conditions within and

seen from the outside. From the inside, changing natural light addresses the notion of temporality by altering one's perception of the contained space. As such, the forms

could be assessed for the nature of their facets and their

ability to create a rich and varied distribution of light in response to varying sun angles throughout the day. Should

the structure be lit from within, at night, it's presence, much like a glowing lantern, can celebrate a civic space for all to enjoy.

'7 / '-A 'C) ;%,~ ~fjj~ \\ 57 .4 -'I

comments & criticisms:

What follows is a partial documentation of the comments and criticisms by my thesis committee and two invited guests, Patricia Patkau of Patkau and Associates in Vancouver and Hugh Stewart from Foster and Partners in London.

Patricia Patkau:

"What was the biggest obstacle in making the jump over from paper to other materials?"

K.G.:

"Paper is extremely thin and therefore very malleable.. .it will distort to permit certain transformation as well as to permit distortion in the final form... meaning certain folded forms would not be able to be folded from a single plane starting point or at all if the units were made of stiff materials such as plywood or sheet metal.

58 PP:

"Were you following one scale as you were working through the different models?" K.G.:

"Yes, I determined this was a small space for a six-ft. person..30',15'." Hugh Stewart:

"It's a fantastic investigation and what it seems you've identified, for me, is that though origami is an attractive proposition at the scale of paper models, it doesn't translate at the scale of real-world building components... what you end up doing is to rigidify it by introducing these struts... and are not using the strength in the folds, as is the case in the paper models.

If this is your area of interest, what I would say you do is to identify what you are trying to achieve and then harness your analytical abilities

to make that happen... because you could go on forever... you need to identify a real task, a real problem and then work backwards as to how you meet the demands imposed by that problem..."

K.G.:

"When working in a smaller scale, felt it was something I physically had a handle on.. .on a larger scale, I must admit that this was not an efficient

deployment of material... I was constantly struggling with the model..needed 3, 4 people to help me hold the nodes in place while I secured the connections...had to go back and forth between modifying what the computer told me was the solution to what reality permitted.."

John Ochsendorf:

"That is one of the key lessons learned -working between the two modes of digital and physical and what hit me was when you said you didn't want to go back to the math.. .how you work between the two. I applaud mentioning the pile of waste material.

H.S.:

"Do you still have a fascination for origami... do you still think there's an application? As an inspiration for form - yes, but I'm less convinced about the construction aspects."

H.S.:

"Struts are hard to engineer.. .you could cover the surface with some splurge and rigidify it like spray-on concrete... the origami could serve as the formwork."

P.P.:

"I appreciate the investigation and how clear you were about your intentions... might recommend you look more carefully at the inhabitants... con-sider daily living... where do you put the furniture and so on?"

Shun Kanda: ₅₉

"What will you do in the next three weeks?"

K.G.:

"Document the work and reflect on the process and its outcome... take a break!" Carol Burns:

"I think there is room for one more wringing out... and one thing I would recommend is this evaluation of tools -one incredible strength of project is that you have gone from your simple fascination of the sophistication of origami, you have bombarded your intuition by way of making with every

kind of tool you could put in your hands... and in this school there a lot of very different tools, right? So with your fabulous friends you made these

fabulous forms... and your thesis could be about the limits of tools available now or a method of probing with a variety of tools... you talked about

the digital and the physical but there is something else that is very strong in your work which is the purely mathematical... which is neither digital nor physical... and I would say that's an area on its own that's worth pursuing.

I'm convinced that you, Ken, could do 8 or 10 more wringing exercises ...just to put a capstone on this investigation..."

J.O.:

Just to wrap up you really covered a full range of things... you worked satisfactorily in the digital world... tried to build them and I applaud you for that... taught us a lot about geometry.. .from this simple form came an astounding richness and complexity of investigation.

bibliography

BOOKS:

Marks,Robert W. "Dymaxion World of Buckminster Fuller", Reinhold Publishing COrporation - NY, 1960 p.178,9 & p.206-8

Fuse,Tomoko "Unit Origami - Multidimensional Transformations", Japan Publications, Inc. -Tokyo 2001

Escrig, F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 2", Computational Mechanics Publications, Southampton 1996

p.63-71

Escrig, F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 3", WIT Press, Southampton 2000

60 p.155-61

Gantes, C.J. "Deployable Structures: Analysis and Design", WIT Press, Southampton 2001 p.3-15,52-6,149

Pearce, Peter "Structure in Nature as a Strategy for Design", MIT Press, Cambridge 1979

p.x-xv, 24-8 JOURNALS:

Praxis vol. 1 "Deceit/fabrication office dA" p.10-13, "Interference Screen" p.24,5 A+U Architecture & Urbanism July-Sept. 1984, p.67-72

CONFERENCE PROCEEDINGS:

Barreto, Paulo Taborda 1992 Lines Meeting on a Surface The "Mars" Paperfolding. in Proc. 3rd Int. Meeting of Origami Science, Mathematics, and Education p.342-59

De Focatiis, D.S.A. & Guest, S.D. 2002 Deployable membranes designed from folding tree leaves Proc. R. Soc. Lond., BE/1-10

http://www-civ.eng.cam.ac.uk/dsl/leaves.pdf

Hall, Judy 1997 Teaching Origami to Develop Visual/Spatial Perception. in Proc. 2nd Int. Conference on Origami Science & Scientific Origami, p.279-91

Kobayashi, H., Kresling, B. & Vincent, J.F.V. 1998 The geometry of unfolding tree leaves. Proc. R. Soc. Lond., p.147-54

61

Kresling, Biruta 1996 Folded and unfolded nature. in Proc. 2nd Int. Conference on Origami Science & Scientific Origami, p.93-107 Ikegami, Ushio 2002 Iso-Area Polyhedra. in Proc. 3rd Int. Meeting of Origami Science, Mathematics, and Education , p.20, 141

Sakoda, James Minoru 1992 Hikari-ori: Reflective Folding. in Proc. 2nd Int. Conference on Origami Science & Scientific Origami, p.333-41 THESES:

Liapi, Katherine A. 2002 Transformable Architecture Inspired by the Origami Art: Computer Visualization as a Tool for Form Exploration. University

WEB SITES: http://www.uspto.gov/patftlindex.html http://kdg.mit.edu/Matrix/matrix.html http://theory.Ics.mit.edu/-edemaine/hypar/ http://www-civ.eng.cam.ac.uk/dsl/leaves.pdf http://www.britishorigami.org.uk/theory/chen.html 62 http://www.greatbuildings.com/buildings/AirForceAcademyChapel.html http://www.greatbuildings.com/architects/FelixCandela.html http://www.bluffton.edu/-sullivanm/bigsail/bigsail.html http://www.americansteelspan.com/storagesheds.html http://www.yupo.com/

*Construction Toy Store "Construction Site" in Waltham -http://www.constructiontoys.com

image credits

* all images are credited

to the author unless listed

in this section (1) (2) (3), (4) (5) (6) (7) (8, 8a) (9) (10) (11-11b) (12) (13) (14,14a) (15, 15a) http://ricegallery.org/view/exhibition/bambooroof.html http://www.Oll.com/lud/pages/architecture/archgallery/ito-serpentine/ Marks,Robert W. "Dymaxion World of Buckminster Fuller'

Escrig, F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 2"

A+U Architecture & Urbanism July-Sept. 1984, p.67-72

Escrig, F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 3" Ikegami, Ushio 2002 Iso-Area Polyhedra

De Focatiis, D.S.A. & Guest, S.D. 2002 Deployable membranes designed from folding tree leaves p.9

Fuse,Tomoko "Unit Origami - Multidimensional Transformations" Gantes, C.J. "Deployable Structures: Analysis and Design" p.54,55

Pearce, Peter "Structure in Nature as a Strategy for Design" p.26,7

http://www.bluffton.edu/-sulivanm/bigsail/bigsail.html http://www.greatbuildings.com/architects/FelixCandela.html http://images.google.com/images

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